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Properties

Label 4-925e2-1.1-c1e2-0-8
Degree $4$
Conductor $855625$
Sign $1$
Analytic cond. $54.5553$
Root an. cond. $2.71774$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·9-s + 6·11-s − 4·16-s + 8·19-s − 4·29-s + 14·41-s − 11·49-s − 8·61-s − 30·71-s + 24·79-s + 16·81-s + 8·89-s + 30·99-s − 10·101-s + 32·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 5/3·9-s + 1.80·11-s − 16-s + 1.83·19-s − 0.742·29-s + 2.18·41-s − 1.57·49-s − 1.02·61-s − 3.56·71-s + 2.70·79-s + 16/9·81-s + 0.847·89-s + 3.01·99-s − 0.995·101-s + 3.06·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(855625\)    =    \(5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(54.5553\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 855625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.916729473\)
\(L(\frac12)\) \(\approx\) \(2.916729473\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad5 \( 1 \)
37$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.41.ao_fb
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \) 2.47.a_bb
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.53.a_adt
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.67.a_es
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.71.be_od
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \) 2.83.a_agb
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.29883822079331656807726094709, −9.648199182049271574205442442906, −9.355469487477673851181236714466, −9.291183712213893153827401128671, −8.860487452846874021455137294794, −8.059017778010705528728782210761, −7.64126389519372121167664886496, −7.21524436642249174260474723064, −7.07383170386774211071998019693, −6.39624914642318759474885510393, −6.13973152908746942882554894416, −5.55410042878517579377978228021, −4.76001955473198647244417968799, −4.54385233423212239601255360530, −4.06303102487752304895733821289, −3.53259304262641880095525664206, −3.03025300731379696849968531194, −2.04390162925547804785771964609, −1.50595709374352737261906253919, −0.918523189452766087992089119907, 0.918523189452766087992089119907, 1.50595709374352737261906253919, 2.04390162925547804785771964609, 3.03025300731379696849968531194, 3.53259304262641880095525664206, 4.06303102487752304895733821289, 4.54385233423212239601255360530, 4.76001955473198647244417968799, 5.55410042878517579377978228021, 6.13973152908746942882554894416, 6.39624914642318759474885510393, 7.07383170386774211071998019693, 7.21524436642249174260474723064, 7.64126389519372121167664886496, 8.059017778010705528728782210761, 8.860487452846874021455137294794, 9.291183712213893153827401128671, 9.355469487477673851181236714466, 9.648199182049271574205442442906, 10.29883822079331656807726094709

Graph of the $Z$-function along the critical line